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These are the user uploaded subtitles that are being translated: 1 00:00:00,000 --> 00:00:02,565 Let's talk about logistic regression, 2 00:00:02,565 --> 00:00:04,500 which is probably the single most 3 00:00:04,500 --> 00:00:07,125 widely used classification algorithm in the world. 4 00:00:07,125 --> 00:00:10,125 This is something that I use all the time in my work. 5 00:00:10,125 --> 00:00:12,390 Let's continue with the example of 6 00:00:12,390 --> 00:00:15,765 classifying whether a tumor is malignant. 7 00:00:15,765 --> 00:00:19,030 Whereas before we're going to use the label 1 or 8 00:00:19,030 --> 00:00:22,540 yes to the positive class to represent malignant tumors, 9 00:00:22,540 --> 00:00:25,710 and zero or no and negative examples 10 00:00:25,710 --> 00:00:27,735 to represent benign tumors. 11 00:00:27,735 --> 00:00:29,760 Here's a graph of the dataset where 12 00:00:29,760 --> 00:00:31,500 the horizontal axis is 13 00:00:31,500 --> 00:00:33,780 the tumor size and 14 00:00:33,780 --> 00:00:37,520 the vertical axis takes on only values of 0 and 1, 15 00:00:37,520 --> 00:00:40,355 because is a classification problem. 16 00:00:40,355 --> 00:00:42,680 You saw in the last video that 17 00:00:42,680 --> 00:00:43,940 linear regression is not 18 00:00:43,940 --> 00:00:45,980 a good algorithm for this problem. 19 00:00:45,980 --> 00:00:50,270 In contrast, what logistic regression we end 20 00:00:50,270 --> 00:00:54,805 up doing is fit a curve that looks like this, 21 00:00:54,805 --> 00:00:58,950 S-shaped curve to this dataset. 22 00:00:58,950 --> 00:01:01,790 For this example, if a patient 23 00:01:01,790 --> 00:01:04,670 comes in with a tumor of this size, 24 00:01:04,670 --> 00:01:07,219 which I'm showing on the x-axis, 25 00:01:07,219 --> 00:01:11,095 then the algorithm will output 0.7 26 00:01:11,095 --> 00:01:13,910 suggesting that is closer or maybe more 27 00:01:13,910 --> 00:01:16,900 likely to be malignant and benign. 28 00:01:16,900 --> 00:01:18,840 Will say more later what 29 00:01:18,840 --> 00:01:22,155 0.7 actually means in this context. 30 00:01:22,155 --> 00:01:28,915 But the output label y is never 0.7 is only ever 0 or 1. 31 00:01:28,915 --> 00:01:32,015 To build out to the logistic regression algorithm, 32 00:01:32,015 --> 00:01:34,850 there's an important mathematical function I like to 33 00:01:34,850 --> 00:01:38,510 describe which is called the Sigmoid function, 34 00:01:38,510 --> 00:01:42,895 sometimes also referred to as the logistic function. 35 00:01:42,895 --> 00:01:46,250 The Sigmoid function looks like this. 36 00:01:46,250 --> 00:01:48,905 Notice that the x-axis of 37 00:01:48,905 --> 00:01:51,685 the graph on the left and right are different. 38 00:01:51,685 --> 00:01:56,405 In the graph to the left on the x-axis is the tumor size, 39 00:01:56,405 --> 00:01:58,390 so is all positive numbers. 40 00:01:58,390 --> 00:02:00,290 Whereas in the graph on the right, 41 00:02:00,290 --> 00:02:02,495 you have 0 down here, 42 00:02:02,495 --> 00:02:06,110 and the horizontal axis takes 43 00:02:06,110 --> 00:02:09,650 on both negative and positive values and have 44 00:02:09,650 --> 00:02:13,130 label the horizontal axis Z. I'm showing 45 00:02:13,130 --> 00:02:17,600 here just a range of negative 3 to plus 3. 46 00:02:17,600 --> 00:02:22,190 So the Sigmoid function outputs value is between 0 and 1. 47 00:02:22,190 --> 00:02:26,075 If I use g of z to denote this function, 48 00:02:26,075 --> 00:02:29,210 then the formula of g of z is equal 49 00:02:29,210 --> 00:02:33,995 to 1 over 1 plus e to the negative z. 50 00:02:33,995 --> 00:02:36,650 Where here e is a mathematical 51 00:02:36,650 --> 00:02:40,195 constant that takes on a value of about 2.7, 52 00:02:40,195 --> 00:02:42,590 and so e to the negative z is that 53 00:02:42,590 --> 00:02:46,000 mathematical constant to the power of negative z. 54 00:02:46,000 --> 00:02:50,705 Notice if z where really be, say a 100, 55 00:02:50,705 --> 00:02:53,690 e to the negative z is e to the 56 00:02:53,690 --> 00:02:57,565 negative 100 which is a tiny number. 57 00:02:57,565 --> 00:03:00,090 So this ends up being 1 58 00:03:00,090 --> 00:03:03,555 over 1 plus a tiny little number, 59 00:03:03,555 --> 00:03:08,330 and so the denominator will be basically very close to 1. 60 00:03:08,330 --> 00:03:11,300 Which is why when z is large, 61 00:03:11,300 --> 00:03:14,090 g of z that is a Sigmoid function 62 00:03:14,090 --> 00:03:17,680 of z is going to be very close to 1. 63 00:03:17,680 --> 00:03:21,470 Conversely, you can also check for yourself 64 00:03:21,470 --> 00:03:25,595 that when z is a very large negative number, 65 00:03:25,595 --> 00:03:30,530 then g of z becomes 1 over a giant number, 66 00:03:30,530 --> 00:03:35,110 which is why g of z is very close to 0. 67 00:03:35,110 --> 00:03:37,520 That's why the sigmoid function has 68 00:03:37,520 --> 00:03:40,250 this shape where it starts very close to 69 00:03:40,250 --> 00:03:46,285 zero and slowly builds up or grows to the value of one. 70 00:03:46,285 --> 00:03:51,830 Also, in the Sigmoid function when z is equal to 0, 71 00:03:51,830 --> 00:03:54,410 then e to the negative z is 72 00:03:54,410 --> 00:03:57,230 e to the negative 0 which is equal to 1, 73 00:03:57,230 --> 00:04:05,445 and so g of z is equal to 1 over 1 plus 1 which is 0.5, 74 00:04:05,445 --> 00:04:10,435 so that's why it passes the vertical axis at 0.5. 75 00:04:10,435 --> 00:04:13,180 Now, let's use this to build up 76 00:04:13,180 --> 00:04:15,595 to the logistic regression algorithm. 77 00:04:15,595 --> 00:04:18,405 We're going to do this in two steps. 78 00:04:18,405 --> 00:04:20,350 In the first step, I hope you 79 00:04:20,350 --> 00:04:23,050 remember that a straight line function, 80 00:04:23,050 --> 00:04:26,170 like a linear regression function can be defined 81 00:04:26,170 --> 00:04:31,205 as w. product of x plus b. 82 00:04:31,205 --> 00:04:34,485 Let's store this value in 83 00:04:34,485 --> 00:04:37,650 a variable which I'm going to call z, 84 00:04:37,650 --> 00:04:39,760 and this will turn out to be the same z 85 00:04:39,760 --> 00:04:41,950 as the one you saw on the previous slide, 86 00:04:41,950 --> 00:04:43,535 but we'll get to that in a minute. 87 00:04:43,535 --> 00:04:47,410 The next step then is to take this value of 88 00:04:47,410 --> 00:04:51,370 z and pass it to the Sigmoid function, 89 00:04:51,370 --> 00:04:53,800 also called the logistic function, 90 00:04:53,800 --> 00:04:56,860 g. Now, g of 91 00:04:56,860 --> 00:05:02,065 z then outputs a value computed by this formula, 92 00:05:02,065 --> 00:05:04,285 1 over 1 plus e to the negative z. 93 00:05:04,285 --> 00:05:07,580 There's going to be between 0 and 1. 94 00:05:07,580 --> 00:05:12,360 When you take these two equations and put them together, 95 00:05:12,360 --> 00:05:17,635 they then give you the logistic regression model f of x, 96 00:05:17,635 --> 00:05:23,290 which is equal to g of wx plus b. 97 00:05:23,290 --> 00:05:27,430 Or equivalently g of z, 98 00:05:27,430 --> 00:05:32,330 which is equal to this formula over here. 99 00:05:32,330 --> 00:05:36,240 This is the logistic regression model, 100 00:05:36,240 --> 00:05:40,240 and what it does is it inputs feature or set 101 00:05:40,240 --> 00:05:44,570 of features X and outputs a number between 0 and 1. 102 00:05:44,570 --> 00:05:47,050 Next, let's take a look at how to 103 00:05:47,050 --> 00:05:50,680 interpret the output of logistic regression. 104 00:05:50,680 --> 00:05:54,710 We'll return to the tumor classification example. 105 00:05:54,710 --> 00:05:57,700 The way I encourage you to think of 106 00:05:57,700 --> 00:06:00,250 logistic regressions output is to think 107 00:06:00,250 --> 00:06:01,630 of it as outputting 108 00:06:01,630 --> 00:06:04,930 the probability that the class or the label 109 00:06:04,930 --> 00:06:10,695 y will be equal to 1 given a certain input x. 110 00:06:10,695 --> 00:06:14,620 For example, in this application, 111 00:06:14,620 --> 00:06:18,320 where x is the tumor size and y is either 0 or 1, 112 00:06:18,320 --> 00:06:20,585 if you have a patient come in 113 00:06:20,585 --> 00:06:23,570 and she has a tumor of a certain size x, 114 00:06:23,570 --> 00:06:26,570 and if based on this input x, 115 00:06:26,570 --> 00:06:29,705 the model I'll plus 0.7, 116 00:06:29,705 --> 00:06:32,210 then what that means is that the model is 117 00:06:32,210 --> 00:06:35,210 predicting or the model thinks there's 118 00:06:35,210 --> 00:06:37,760 a 70 percent chance that the true label 119 00:06:37,760 --> 00:06:40,855 y would be equal to 1 for this patient. 120 00:06:40,855 --> 00:06:43,070 In other words, the model is telling 121 00:06:43,070 --> 00:06:45,605 us that it thinks the patient has 122 00:06:45,605 --> 00:06:47,660 a 70 percent chance of 123 00:06:47,660 --> 00:06:50,765 the tumor turning out to be malignant. 124 00:06:50,765 --> 00:06:53,525 Now, let me ask you a question. 125 00:06:53,525 --> 00:06:56,105 See if you can get this right. 126 00:06:56,105 --> 00:07:00,695 We know that y has to be either 0 or 1, 127 00:07:00,695 --> 00:07:04,400 so if y has a 70 percent chance of being 1, 128 00:07:04,400 --> 00:07:07,710 what is the chance that it is 0? 129 00:07:07,730 --> 00:07:11,305 So y has got to be either 0 or 1, 130 00:07:11,305 --> 00:07:13,670 and thus the probability of it being 131 00:07:13,670 --> 00:07:16,400 0 or 1 these two numbers 132 00:07:16,400 --> 00:07:20,095 have to add up to one or to a 100 percent chance. 133 00:07:20,095 --> 00:07:22,940 That's why if the chance of y being 134 00:07:22,940 --> 00:07:25,805 1 is 0.7 or 70 percent chance, 135 00:07:25,805 --> 00:07:28,580 then the chance of it being 0 has got to 136 00:07:28,580 --> 00:07:31,990 be 0.3 or 30 percent chance. 137 00:07:31,990 --> 00:07:33,800 If someday you read 138 00:07:33,800 --> 00:07:35,510 research papers or blog pulls 139 00:07:35,510 --> 00:07:37,055 of all logistic regression, 140 00:07:37,055 --> 00:07:40,220 sometimes you see this notation that f 141 00:07:40,220 --> 00:07:43,190 of x is equal to p of 142 00:07:43,190 --> 00:07:46,280 y equals 1 given 143 00:07:46,280 --> 00:07:50,990 the input features x and with parameters w and b. 144 00:07:50,990 --> 00:07:53,810 What the semicolon here is used to 145 00:07:53,810 --> 00:07:56,900 denote is just that w and b are 146 00:07:56,900 --> 00:08:00,800 parameters that affect this computation of what is 147 00:08:00,800 --> 00:08:02,840 the probability of y being equal to 1 148 00:08:02,840 --> 00:08:05,725 given the input feature x? 149 00:08:05,725 --> 00:08:07,130 For the purpose of this class, 150 00:08:07,130 --> 00:08:08,450 don't worry too much about what 151 00:08:08,450 --> 00:08:11,810 this vertical line and what the semicolon mean. 152 00:08:11,810 --> 00:08:14,120 You don't need to remember or 153 00:08:14,120 --> 00:08:16,820 follow any of this mathematical notation for this class. 154 00:08:16,820 --> 00:08:18,290 I'm mentioning this only 155 00:08:18,290 --> 00:08:20,770 because you may see this in other places. 156 00:08:20,770 --> 00:08:23,465 In the optional lab that follows this video, 157 00:08:23,465 --> 00:08:24,980 you also get to see how 158 00:08:24,980 --> 00:08:28,225 the Sigmoid function is implemented in code. 159 00:08:28,225 --> 00:08:30,300 You can see a plot that uses 160 00:08:30,300 --> 00:08:32,420 the Sigmoid function so as to do 161 00:08:32,420 --> 00:08:34,550 better on the classification tasks 162 00:08:34,550 --> 00:08:36,820 that you saw in the previous optional lab. 163 00:08:36,820 --> 00:08:39,350 Remember that the code will be provided to you, 164 00:08:39,350 --> 00:08:41,345 so you just have to run it. 165 00:08:41,345 --> 00:08:45,185 I hope you take a look and get familiar with the code. 166 00:08:45,185 --> 00:08:47,615 Congrats on getting here. 167 00:08:47,615 --> 00:08:51,534 You now know what is the logistic regression model 168 00:08:51,534 --> 00:08:53,530 as well as the mathematical formula 169 00:08:53,530 --> 00:08:55,945 that defines logistic regression. 170 00:08:55,945 --> 00:08:57,505 For a long time, 171 00:08:57,505 --> 00:09:00,670 a lot of Internet advertising was actually driven 172 00:09:00,670 --> 00:09:04,390 by basically a slight variation of logistic regression. 173 00:09:04,390 --> 00:09:07,160 This was very lucrative for some large companies, 174 00:09:07,160 --> 00:09:08,560 and this is basically the algorithm 175 00:09:08,560 --> 00:09:10,000 that decided what ad was 176 00:09:10,000 --> 00:09:13,604 shown to you and many others on some large websites. 177 00:09:13,604 --> 00:09:15,480 Now, there's, even more, 178 00:09:15,480 --> 00:09:17,155 to learn about this algorithm. 179 00:09:17,155 --> 00:09:18,580 In the next video, 180 00:09:18,580 --> 00:09:22,180 we'll take a look at the details of logistic regression. 181 00:09:22,180 --> 00:09:24,850 We'll look at some visualizations and also 182 00:09:24,850 --> 00:09:28,280 examines something called the decision boundary. 183 00:09:28,280 --> 00:09:30,650 This will give you a few different ways to 184 00:09:30,650 --> 00:09:33,515 map the numbers that this model outputs, 185 00:09:33,515 --> 00:09:36,170 such as 0.3, or 0.7, 186 00:09:36,170 --> 00:09:42,440 or 0.65 to a prediction of whether y is actually 0 or 1. 187 00:09:42,440 --> 00:09:44,990 Let's go on to the next video to learn 188 00:09:44,990 --> 00:09:48,300 more about logistic regression.13661

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